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Gibbs' papers defined what Gibbs called the "thermodynamic surface," which expressed the relationship between the volume, entropy, and energy of a substance at different temperatures and pressures. However, Gibbs did not include any diagrams of this surface.[4][7] After receiving reprints of Gibbs' papers, Maxwell recognized the insight afforded by Gibbs' new point of view and set about constructing physical three-dimensional models of the surface.[8] This reflected Maxwell's talent as a strong visual thinker[9] and prefigured modern scientific visualization techniques.[4]

Maxwell sculpted the original model in clay and made three plaster casts of the clay model, sending one to Gibbs as a gift, keeping the other two in his laboratory at Cambridge University.[4] Maxwell's copy is on display at the Cavendish Laboratory of Cambridge University,[4][10] while Gibbs' copy is on display at the Sloane Physics Laboratory of Yale University,[11] where Gibbs held a professorship. A number of historic photographs were taken of these plaster casts during the middle of the twentieth century – including one by James Pickands II, published in 1942[1] – and these photographs exposed a wider range of people to Maxwell's visualization approach.

Maxwell's sketch of the isothermals and isopiestics on his surface (Plate IV of his collected letters).

Maxwell drew lines of equal pressure (isopiestics) and of equal temperature (isothermals) on his plaster cast by placing it in the sunlight, and "tracing the curve when the rays just grazed the surface."[3] He sent sketches of these lines to a number of colleagues.[12] For example, his letter to Thomas Andrews of 15 July 1875 included sketches of these lines.[3] Maxwell provided a more detailed explanation and a clearer drawing of the lines in the revised version of his book Theory of Heat,[13] and a version of this drawing appeared on a 2005 US postage stamp in honour of Gibbs.[7]

As well as being on display in two countries, Maxwell's model lives on in the literature of thermodynamics, and books on the subject often mention it,[14] though not always with complete historical accuracy. For example, the thermodynamic surface represented by the sculpture is often reported to be that of water,[14] contrary to Maxwell's own statement.[3]

Around 1900, the Dutch scientist Heike Kamerlingh Onnes, together with his student Johannes Petrus Kuenen and his assistant Zaalberg van Zelst, continued Maxwell's work by constructing their own plaster thermodynamic surface models.[16] These models were based on accurate experimental data obtained in their laboratory, and were accompanied by specialised tools for drawing the lines of equal pressure.[16]

^ abcdMaxwell, James Clerk (1995-01-01). Maxwell on Heat and Statistical Mechanics: On "Avoiding All Personal Enquiries" of Molecules. p. 248. ISBN9780934223348. I think you know Prof. J. Willard Gibbs's (Yale College Connecticut) graphical methods in thermodynamics. Last winter I made several attempts to model the surface which he suggests, in which the three coordinates are volume, entropy and energy. The numerical data about entropy can only be obtained by integration from data which are for most bodies very insufficient, and besides it would require a very unwieldy model to get all the features, say of CO2, well represented, so I made no attempt at accuracy, but modelled a fictitious substance, in which the volume is greater when solid than when liquid; and in which, as in water, the saturated vapour becomes superheated by compression. When I had at last got a plaster cast I drew on it lines of equal pressure and temperature, so as to get a rough motion of their forms. This I did by placing the model in sunlight, and tracing the curve when the rays just grazed the surface... I send you a sketch of these lines..." (letter to Thomas Andrews, 15 July 1875)

^Maxwell and Harman, pp. 230-231: "I enclose a rough sketch of the lines on Gibbs' surface, co-ordinates Volume Entropy Energy in an imaginary substance in which the principal features of known substances can be represented on a convenient scale." (letter to James Thomson, 8 July, 1875)